Chapter 6 Frequency Response, Bode Plots and Resonance Signal Processg is concerned with manipulatg Signals to extract Inormation and to use that ormation to generate other useul Signals
Goal. Fundamental Concepts o Fourier Analysis. 2. Filter o a given put consistg o susoidal components usg the ilter s transer unction. 3. Low-pass or High-pass ilter & Transer Functions. 4. Decibels, Logarithmic Frequency Scales, and Bode plots. 5. Understand arious Filters 6. Simple Digital Signal Processg System
Fourier Analysis All Real-World Signals are Lear Sums o Susoidal components havg various requencies, amplitudes & phases. τ ω 2π ] s cos [ 2 ) ( t n b t n a a t n n n o ω ω + + + τ ω τ o o t t n dt t n t a cos ) ( 2 + τ ω τ o o t t n dt t n t b s ) ( 2
Even Step unction can be analyzed
Fourier Analysis o Music Time t a n, b n Let ω 2 π/ τ a t a n t b nωt () + ncos ω + ns 2 n 2 to + τ an dt ()cos t nωt τ t 2 to + τ bn dt ()s t nωt τ t F(ω) or (t) A kd o Spectrum Analyzer
Filters Goal : Reta & Reject the Components Certa Frequency ranges Filters process the Susoid Components o an put signal dierently dependg o the requency o each component. Code as Political Filter!
Filters Types Example o Ideal Low Pass ilter
Transer Functions Transer unction H( ) o Two-Port Filter : Phasors Output / Phasors Input as a unction o requency H H ( ) ( ) H ( ) : Amplitude Change o Each Frequency Component : Phase Change o Each Frequency Component H ( ) H ( )
Filter o Multiple Components. Determe requency & Phasors or each put component. 2. Determe Complex Transer unction or each component. 3. Obta the phasors or each put component by multiplyg the phasors or each put component by transer-unction 4. Convert the phasors or each put components to time unctions. Add these time unctions to produce the put.
Experimental Determation o H() - Application o Susoidal Source at ixed requency - Measurement o Output Signal Amplitude, Phase - Procedure is repeated or each requency o terest
Active Noise Canceller Noise Reduction Airplane or Car - Passive : Absorber - Active : Canceller Measurement o Signal - Source : Enge Inverse Signal to Loud Speaker - Receiver : Seat Best Cancellation to Passenger
Low Pass Filter
First-Order Low Pass Filters Total Impedance : Z t Z t Z R + Z C R + R + jωc j2πc I Z t R + j2πc IZ C j2πc I j2πc R + j2πc H ( ) + j2πrc
First-Order Low Pass Filters H ( ) + j ( ) B B 2πRC H ( ) + ( ) ( ) 2 B B H arctan
Low Pass RC & RL Filters H ( ) + j ( ) B B 2πRC I B R 2πL B
Decibels H ( ) log H ( ) db 2 Clear Description o Frequency o Certa Range o reqeuncy
Cascaded Two-Port Networks H ( ) 2 2 2 H ( ) H ( ) H ( ) 2 ( ) H ( ) H ( ) db db 2 db H +
Logarithmic Frequency Scales On a logarithmic scale, the variable is multiplied by a given actor or equal crements o length along the axis. Decade : a range o requencies or which the ratio o the highest requency to the lowest is. 2 Number o Decades log Octave : a two-to-one change requency or clear description o low range requency such as 2 to 2 MHz. ( 2 ) ( 2) 2 log Number o Octaves log 2 log
Bode Plots Magnitude Decibels versus a logarithmic scale requency. H ( ) log H ( ) db 2 vs log In Low-Pass ilter H ( ) + j ( ) B H ( ) log db + B 2 H ( ) db B Corner requency Breakg Frequency ( ) 3dB at B H db H ( ) 2log db B
Phase Bode Plot. A horizontal le at zero or < B /. 2. A slopg le rom zero phase at B / to 9 at B. 3. A horizontal le at 9 or > B.
Example o Bode Plot
First-Order High-Pass Filters Total Impedance : Z t I Z t Z R + j2πc t Z R + Z C R + R + jωc j2πc IZR RI R R + j2πc H ( ) j2πrc + j2πrc
H H ( ) ( ) High Pass RC Filters j + ( ) B j( ) B B 2πRC H + ( ) ( ) 9 o arctan 2 B B
High Pass RC & RL Filters B 2πRC j2πl B R 2πL B
Bode Plot o High Pass ilter
C Summary High Pass Filter Low Pass Filter ( / C ) ( / ) [ ] 2 + / 2 C [ ( ) ] 2 + / / 2 Red Le : Multiplication 2 C ( / C ) ( / ) [ + 2 ] C Red Le shows Band Selection! ( 2 ) c / πrc
Band Pass RC Filters with Cascade Network 2 2 ( ) + + ( + r) H L H 2 [ ] 2 L H H 2πR C L L L 2πR C H H r R R H L Low Pass Filter High Pass Filter L L : Pass Band Response o a band-pass ilter
Band Reject RC Filters L 2πR C H H H 2πR C L L Not eective combation Use Active ilter! C
RC Circuit as Dierentiator oltage oltage across across C : R : I d C [ ] dt R For small RC RC d [ ] dt d dt d dt C d dt >> R RC d dt Dierentiation o Input oltage
Dierentiation o Square Wave Leadg Edge Detector
RC Circuit as Integrator oltage oltage across across R : C : I C d dt R For Large RC RC d [ ] dt >> C d dt R dt + const RC Integration o Input oltage
Mechanical Resonance Electrical Resonance Wd & Bridge Electron & RLC
Series Resonance Z s ( ) j2πl + R j 2πC Resonance Frequency 2π LC Quality actor 2π L Q s R 2π CR Total Impedance Z s ( ) R + jq s R L C
Impedance Characteristics
Filter Case o Series Resonance R Band Pass L High Pass Band Notch C Low Pass
Second-Order Low Pass Filter Z s Z s ( ) j2πl + R j 2πC ( ) R + jq s 2π R L 2π CR Q s 2π LC Capacitor Component H ( ) ( s ) ( ) jq + jq s
First & Second Order Low Pass Filters
Second-Order High Pass Filter Z s ( ) j2πl + R j 2πC Z s ( ) R + jq s 2π R L 2π CR Q s 2π LC Inductor Component H ( ) ( ) s ( ) jq + jq s
Bode Plot o Second-Order High Pass Filter
Series Resonant Circuit as a Band-Pass Filter + Z s ( ) j2πl + R j 2πC - Z s ( ) R + jq s 2π R L 2π CR Q s 2π LC Resistor Component + jq s ( )
Bode Plot o Second-Order Band Pass Filter B H L Qs Q s >> H + B 2 L B 2
Series Resonant Circuit as Band-Reject Filter Z Z s LC ( ) R + jq Z L + Z C s j[2πl ] 2πC 2π R L 2π CR Q s 2π LC ( ) ( ) jqs + jq s
Bode Plot o Second-Order Band Reject Filter
PARALLEL RESONANCE Quality actor Z p Total Impedance ( R) + j2πc j( 2πL) Q R 2π CR p 2π L Z p R + jq p Resonance requency ( ) 2π LC
Parallel Resonant Circuit as Band-Pass Filter IR + jq p ( ) B H L Qp
Resonance Filter LC o π 2 / ] 2 [2 2 2 L C j j C L j C L LC π π π π + Z Z Z LC R total Z Z Z + RC Q o 2π C L j R total π 2π ) 2 / ( + Z ( ) jq s LC R LC + Z Z Z
Photograph o arious Noise Filter
SAW Filter SAW : Surace Acoustic Wave o
DIGITAL SIGNAL PROCESSING Analog-Digital Converter DSP Digital-Analog Converter ADC : Conversion o Signals rom Analog to Digital Form s > 2 : Samplg Rate (requency) H I a signal contas no components with requencies higher than H, the signal can be exactly reconstructed rom its samples, provided that the samplg rate s is selected to be more than twice H.
Example o ADC (3bit ADC) Samplg Error T s k N 2 Quantization Error
Digital Low Pass Filter KCL at Top Node y ( t) R x( t) + C dy( t) dt y () t with y x dy( t) t) + τ dt τ RC ( ( t) τ T a + τ T ( n) ay( n ) + ( a) x( n) dy δy( t) y( n) y( n ) dt δ t T
Application o Digital Low Pass ilter to Step Signal Step Signal (Red Le) o : Digitized Signal x : Filtered Signal Step unction must be smoothed by Low Pass ilter, leadg to Constant value y τ T a + τ T ( n) ay( n ) + ( a)() x n
Signal with Typical Noise
Filtered Signal